Let $\{A_i\}$ be a net of hermitian operators on a separable Hibert space $\mathbb{H}$ and suppose that there is a hermitian operator T such that $A_{i}\le T$ for all i. If $\{<A_i h,h>\}$ is an increasing net in $\mathbb{R}$ for every $h$ in $\mathbb{H}$, then there is a hermitian operator $A$ such that $A_i\rightarrow A (WOT)$ in weak operator topology.
Any help would be appreciated.
This is like Lemma 5.1.4 in Kadison & Ringrose. The key is to use the weak-operator compactness of the unit ball of $\mathcal{B}(\mathbb{H})$, Theorem 5.1.3 (p. 306). You then use the general topological fact that compactness is equivalent to "every net has a convergent subnet." Alas, I'm not sure I can shed much light on the proof of compactness ... hopefully you can find that theorem in Kadison & Ringrose or a similar reference.